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文库首页行业教育图像处理与分析：lec5 Image Enhencement in Frequency Domain(I).pdf

图像处理与分析：lec5 Image Enhencement in Frequency Domain(I).pdf

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图像处理与分析：lec5 Image Enhencement in Frequency Domain(I).pdf

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Image Processing and Analysis

Lecture 5!Image Enhencement in Frequency Domain(I)

W.Q. Wang (SCCE,UCAS) Image Processing and Analysis November 10, 2015 1 / 42

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Outline

2-D Discrete Fourier Transform

Filtering in the Frequency Domain

Obtaining Frequency Domain Filters from Spatial Filters

Generating Filters Directly in the Frequency Domain

W.Q. Wang (SCCE,UCAS) Image Processing and Analysis November 10, 2015 2 / 42

2-D Discrete Fourier Transform

2-D Fourier Transform

Any function that periodically repeats itself can be expressed as the

sum of sines and/or cosines of diﬀerent frequencies, each multiplied

by a diﬀerent coeﬃcient (Fourier series).

Even functions that are not periodic (but whose area under the curve

is ﬁnite) can be expressed as the integral of sines and/or cosines

multiplied by a weighting function (Fourier transform).

The frequency domain refers to the

plane of the two dimensional discrete

Fourier transform of an image.

The purpose of the Fourier transform is

to represent a signal as a linear

combination of sinusoidal signals of

various frequencies.

W.Q. Wang (SCCE,UCAS) Image Processing and Analysis November 10, 2015 3 / 42

2-D Discrete Fourier Transform

2-D Continuous Fourier Transform

The one-dimensional Fourier transform and its inverse

Fourier transform

F (u) =

∞

−∞

f(x)e

−j2πux

dx, where j =

√

−1

Inverse Fourier transform:

f(x) =

∞

−∞

F (u)e

j2πux

du e

jθ

= cos θ + j sin θ

The two-dimensional Fourier transform and its inverse

Fourier transform

F (u, v) =

∞

−∞

∞

−∞

f(x, y)e

−j2π(ux+vy)

dxdy

Inverse Fourier transform:

f(x, y) =

∞

−∞

∞

−∞

F (u, v)e

j2π(ux+vy)

dudv

W.Q. Wang (SCCE,UCAS) Image Processing and Analysis November 10, 2015 4 / 42

2-D Discrete Fourier Transform

The one-dimensional Discrete Fourier transform (DFT) and its inverse

Fourier transform

F (u) =

M−1

x=0

f(x)e

−

j2πux

for u = 0, 1, 2, ..., M −1

Inverse Fourier transform:

f(x) =

M−1

u=0

F (u)e

j2πux

for x = 0, 1, 2, ..., M −1

Since e

jθ

= cos θ + j sin θ, then DFT can be redeﬁned as

F (u) =

M−1

x=0

f(x)[cos

2πux

− j sin

2πux

]

for u = 0, 1, 2, ..., M −1

Frequency (time) domain: the domain (values of u) over which the

values of F(u) range; because u determines the frequency of the

components of the transform.

Frequency (time) component: each of the M terms of F (u).

W.Q. Wang (SCCE,UCAS) Image Processing and Analysis November 10, 2015 5 / 42

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